Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of TakeutiZaring p. 76. (Contributed by NM, 2-Sep-2003) (Revised by Mario Carneiro, 10-Sep-2013)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r1suc | |- ( A e. On -> ( R1 ` suc A ) = ~P ( R1 ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r1sucg | |- ( A e. dom R1 -> ( R1 ` suc A ) = ~P ( R1 ` A ) ) |
|
| 2 | r1fnon | |- R1 Fn On |
|
| 3 | 2 | fndmi | |- dom R1 = On |
| 4 | 3 | eqcomi | |- On = dom R1 |
| 5 | 1 4 | eleq2s | |- ( A e. On -> ( R1 ` suc A ) = ~P ( R1 ` A ) ) |